A STUDY  OF  SUCCESS  IN  AN  INTRODUCTORY 
EXAMINATION  IN  HIGH  SCHOOL  ALGEBRA 
AS  RELATED  TO  SUCCESS  IN 
COLLEGE  ALGEBRA 


By 

SARAH  HELEN  TAYLOR 

A.  B.  Illinois  College,  1920 


THESIS 


SUBMITTED  IN  PARTIAL  FULFILLMENT  OF  THE  REQUIREMENTS 
FOR  THE  DEGREE  OF  MASTER  OF  ARTS  IN  MATHEMATICS 
IN  THE  GRADUATE  SCHOOL  OF  THE 
UNIVERSITY  OF  ILLINOIS,  1922 


URBANA,  ILLINOIS 


J 


UNIVERSITY  OF  ILLINOIS 
THE  GRADUATE  SCHOOL 


. — v.'  0 1 92_£- 

I HEREBY  RECOMMEND  THAI'  THE  THESIS  PREPARED  UNDER  MY 
SUPER VI SION  B Y laraU  Eel ?.i  Caylor , . 

ENTITLED A_ Study  of  Success in  an  Introductory  u>y;y  y.  __ 

in  Hi  ;i  School  Al-oErr  m Related  to  Success  in  dol  Ip-p 

Algebra 

BE  ACCEPTED  AS  FULFILLING  THIS  PART  OF  THE  REQUIREMENTS  FOR 
THE  DEGREE  OF  "r  3 c r o f / : i y 


Recommendation  concurred  in* 


Committee 

on 

Final  Examination* 


•Required  for  doctor's  degree  but  not  for  master’s 


Digitized  by  the  Internet  Archive 
in  2016 


https://archive.org/details/studyofsuccessinOOtayl 


Contents 


(a) 


Chapter  I. 

The  Preliminary  Test. 

Page 

Introduction 1 

The  Questions 2 

Method  of  Grading 5 

Distribution  of  Grades 6 

Average  or  Mean 7 

Median 8 

Quart ile  Deviation 8 

Influence  of  the  Class  Interval  on  the  Mean 8 

Stanuard  Deviation 11 

Histogram  and  Probability  Curve 13 

Errors 16 

Conclusions 18 

Chapter  II. 

The  Final  Examination. 

The  Questions 19 

Distribution  of  Grades £0 

Mean,  Median  and  Stanuard  Deviation 22 

Probability  Curve 24 

Discussion  of  Semester  Grades 26 


Influence  of  the  Examination  on  the  Semester  Grace.... 


28 


Chapter  III 


Co) 


Correlation  of  Grades  of  the  Two  Examinations. 


Page 

Correlation  Defined.. 30 

Correlation  Table 30 

Coefficient  of  Correlation 31 

Probable  Error 32 

Regression  Lines 32 

Correlation  Ratios 33 

blakeman  Test 34 

Scatter  Diagram 35 

Conclusions  and  Suggestions 41 

Chapter  IV. 

C orrelation  with  Respect  to  the  Time  Element . 

Time  Element 46 

Correlation  toy  Years 47 

Discussion  of  and  Values 52 

Conclusions 53 

Summary 54 

bibliography 56 


1 


Chapter  I. 

The  Preliminary  Examination. 

Introduction . 

The  examination  which  furnished  the  data  for  this  investiga- 
tion was  given  in  a two  and  one  half  hour  period,  early  in 
October,  1921,  to  those  students  taking  the  three  hour  course  in 
college  algebra  at  the  University  of  Illinois.  The  prerequisite 
for  this  course  is  one  and  one  half  years  of  high  school  algebra 
and  one  year  of  plane  geometry.  Preceding  this  examination,  the 
classes  had  eight  review  lessons  over  high  school  algebra,  each 
instructor  following  the  outlined  review  suggested  by  the  depart- 
ment which  is  here  included.  The  text  used  was  Bietz  and  Cra- 
thorne . 

Lesson  I.  Introductory  lecture.  Explain  review  and  test. 
Give  an  analysis  of  high  school  algebra  with  emphasis  placed  on 
tractions . 


Lesson 

II. 

Theory  of 

exponent  s , 

sections  6- 

12. 

Lesson 

III. 

. Parenthe 

sis,  complex 

fractions , 

sections 

13-15. 

Lesson 

IV. 

Pact or ing 

, radicals, 

sections  16 

-18. 

Lesson 

V. 

Radicals , 

sections  19- 

20 . 

Lesson 

VI. 

Equat ions 

ana  iaentit 

ies.  Solut 

ions  of  s 

imp  1 e 

. ions  , £ 

sect  i 

Lons  30-35. 

Lesson  VII.  Elimination  by  addition,  subtraction,  and 
substitution,  (without  determinants). 


. 

■ 


. 


’ 


2 


The  _2ue  §,t  i ons . 

The  examination  was  made  up  through  the  cooperation  of  the 
whole  staff.  In  the  spring  of  1921,  a letter  was  sent  to  each  of 
the  professors  and  instructors  who  had  classes  in  algebra  asking 
for  a list  of  fifteen  problems  suitable  for  an  examination  over 
high  school  work.  With  each  problem  the  instructor  was  asked  to 
state  the  principle  on  which  the  problem  examined  so  that  each 
problem  would  involve  a single  item , --definition , process,  princi- 
ple or  ability, — rather  than  a complex  item.  Two  members  of  the 
department  who  are  especially  interested  in  the  teaching  of 
Freshman  Mathematics,  listed  twenty-five  principles  which,  in 
their  judgment,  the  examination  should  cover  and  then  chose  from 
the  lists  submitted  by  the  instructors  the  problem  which  best  tested 
each  principle.  Questions  involving  single  principles  are  most 
suitable  for  test  use  in  that  a student  can  classify  and  work  such 
problems  in  the  short  time  allowed  for  the  test  much  more  readily 
than  he  could  think  through  and  solve  problems  involving  complex 
principles.  A student  may  be  able  to  handle  all  the  algebraic 
processes  singly  but  not  be  able  to  connect  them  by  solving  a 
problem  which  involves  several  of  them  and  so  he  will  completely 
fail  on  the  problem  just  as  does  the  student  who  can  use  them 
neither  singly  or  together.  The  wide  difference  in  student  ability 
to  use  algebraic  processes  shows  up  much  more  clearly  when  questions 
involving  single  rather  than  complex  principles  are  used. 

The  resulting  list  of  twenty-five  questions  is  given  below. 

The  examination  included  no  problem  in  quadratics  simply  because 
the  review  period  was  too  short  to  include  this  subject,  and  it 


. 


. 


. 


3 


was  left  for  a subsequent  review.  The  examination  was  the  result 
of  a thoughtful  and  cooperative  effort  of  the  teaching  staff.  On 
careful  inspection,  one  nay  question  the  length  of  the  examination, 
the  difficulty  or  simplicity  of  certain  questions  or  the  correct- 
ness of  emphasis  on  certain  types  of  problem.  However,  a criticism 
of  the  questions  can  better  be  considered  at  the  end  of  the  chapter 
when  the  purpose  and  possible  uses  of  such  an  examination  are  dis- 


cussed. 


MATHEMATICS  3. 


October,  1921. 


Answer  all  questions. 

1.  Write  the  following  English  statements  in  algebraic  language: 

(a)  The  perimeter  of  a square  equals  four  times  a side. 

(b)  The  area  of  a square  is  equal  to  the  square  of'  a side. 

3 2 2 3 2 o ' ; |7 

2.  If  p=x  + 3x  y + 3xy  t y and  Q = 2x  + 5 x^y  - 4xy^  - y 
find  2P  - q. 

3.  What  should  be  considered  as  negative  if  the  following  are 
considered  as  positive: 

(a)  west  longitude. 

(b>  dollars  gain. 

(c)  miles  northeast. 

(d>  cubic  inches  of  expansion. 

(e)  excess  of  water  pumped  from  a well  over  that  flowing  into 
it  in  the  same  time. 

i 2 

4.  s - ■£  gt  t at  , in  which  g =.  16. 8 , t=lO,  a =65.  3.  Find  s . 

5.  Obtain  an  expression  equal  to  (x(l-x)~  - x (x-1) (x+lV) x 

3 2 2 

6.  Factor  x -2xy+  yw-  z . 

7.  Factor  ax  + ay  -t-bx  +by. 

2a 

8.  Show  that  2b(b-a)  = - — — — — Give  a reason  for  each  step. 

- b (a-b  ^ . 


. 


. 

. 


■ 

. 


. 


. 


9.  Simplify  (h  + o-a  > ("b  - a^° D ) ( ) 

° + a 

10.  Seduce  to  a simple  fraction: 

a- x _ a+x 

a+x a-x 

a-x  a+x 
a+x  a-x 


11.  Solve 

and  check  3(2x-l)  - 

4 (6x-5)  = 

H* 

to 

X 

1 

5)  - 22. 

12.  Define 

and  illustrate  the 

solution 

of  an 

equation. 

13.  Solve 

for  x:  2x+3  2x-3 

48 

2x-3  — 2x-+3 

4x2-9 

~ u 

Check  the  result. 

14.  Solve  (x-+5y)  = 

(3x-lly)  = 95 

15.  State  three  laws  of  exponents  and  prove  one  for  positive 
integers . 

16.  Find  the  value  of  12^  - 4^  + 3 ^+  0^  - 8 \ 

17.  Simplify  and  write  with  positive  exponents. 


18.  Simplify  3VT47  - 3 vT 

3 

IS.  Add:  VT  + Y27  + 3VT8  - Y?6  + W. 

30.  Simplify  7 +■  /2  7 _ yfT 

vT“  v—n#" 

21.  A man  of  4o  has  a son  lo  years  old;  in  how  many  years  will 
the  father  he  three  times  as  old  as  the  son? 

22.  A square  grass  plot  would  contain  73  sq.  ft.  more  if  each 
side  were  one  foot  longer.  Find  the  length  of  one  side  of  the 
plot . 

23.  Which  of  the  following  equalities  are  true  for  all  values  of 
x and  which  are  true  for  only  one  value: 


. 


. 


. 

■ 


' 


. 

~ VCM)  *'  fat  I 

. 


. 


: 


. 

. 


. 


: 


5 


3x^1  = x 4-2x  t-1, 
x + 4x  = 5 

X - X 


2x  -7  - 6 

(x+2)  (x- 2)  ss  x2  - 4 


24.  (a+b) 2 - 

2 2 

a +■  2ab  + b . 

State 

this 

law  in  your  own  words. 

3 

4 

25.  Which  is 

the  largest? 

'f*. 

1~7. 

Why? 

Method  of  Grading . 

The  examination  was  graded  in  committee;  twenty-five  members 
of  the  staff  working  at  the  same  time  so  that  each  instructor 
corrected  only  one  and  the  same  question,  on  each  paper.  No  grade 
was  placed  on  the  question;  instead  the  instructor  put  a long  dash 
below  the  problem  he  graded  in  order  that  the  next  person  might 
rind  the  place  more  readily,  and  then  turned  the  book  over  and 
placed  the  grade  for  that  problem  on  the  back  cover.  The  grades 
on  the  back  were  written  in  a column  in  the  order  in  which  the 
questions  occurred.  The  method  of  committee  grading  resulted  in 
greater  accuracy  and  less  variation,  it  is  thought.  ±$y  such  a 
method  each  one  has  to  keep  in  mind  only  one  problem  and  the 
evaluation  or  dirrerent  types  or  mistakes  in  that  problem.  The 
ract  that  no  grades  were  placed  on  the  questions  themselves  tenuea 
to  do  away  with  any  suggestion  or  ability  or  lack  of  ability 
which  might  influence  the  instructor  in  his  grading  ir  he  saw 


numerous  fours  or  zeros  on  the  preceding  problems. 

In  grading  the  papers,  each  problem  was  considered  to  be  of 


. 


. 


6 


the  same  value  as  any  other.  The  problems  were  inarmed  on  a per- 
centage basis,  each  one  correctly  solved  counting  four.  Ir  the 
algebraic  process  involved  was  correctly  stated  and  used  but  a 
mistake  was  made  in  some  arithmetic  process,  three-fourths  credit 
was  allowed  and  a grade  of  three  given  on  the  problem.  Since  some 
problems  had  two  and  some  four  parts,  grades  of  zero,  one,  two, 
three  and  four  were  possible  on  practically  every  problem. 

The  distribution  ox  grades  for  the  nine  hundred  and  seventy- 
seven  students  taking  this  examination  is  shown  in  the  following 
frequency  table: 


Table 

showing  number  or  students  receiving  each 

grade . 

Grade 

Frequency 

Gr  ade 

Frequency 

Grade 

Frequency 

100 

1 

84 

28 

68 

17 

99 

0 

83 

17 

67 

22 

98 

4 

82 

15 

66 

26 

97 

3 

81 

22 

65 

11 

96 

2 

80 

26 

64 

21 

95 

6 

79 

20 

63 

28 

94 

8 

78 

21 

62 

23 

93 

13 

77 

22 

61 

17 

92 

6 

76 

16 

60 

14 

91 

13 

7 5 

25 

59 

18 

90 

14 

74 

26 

58 

14 

89 

11 

73 

16 

57 

18 

88 

16 

72 

20 

56 

16 

87 

14 

71 

16 

55 

8 

86 

18 

70 

17 

54 

8 

85 

14 

69 

27 

53 

9 

. 


7 


Grade 

Frequency 

Grade 

Fre  quency 

52 

16 

31 

3 

51 

16 

30 

5 

50 

13 

29 

2 

49 

14 

28 

6 

48 

13 

27 

2 

47 

10 

26 

2 

46 

9 

25 

0 

45 

9 

24 

3 

44 

14 

23 

1 

43 

12 

22 

1 

42 

8 

21 

0 

41 

2 

20 

3 

40 

3 

19 

1 

39 

9 

18 

1 

38 

5 

17 

1 

37 

11 

16 

1 

36 

6 

15 

0 

35 

11 

14 

2 

34 

7 

13 

1 

33 

3 

12 

1 

32 

2 

11 

1 

Total977 

1 

The  Mean. 


The  average  or  arithmetic  mean  is  obtained  by  finding  the 
sum  of  the  numerical  values  of  the  measures  multiplied  by  the 

1.  Rugg,  Statistical  Methods,  p.  117 


8 


frequency  of  their  occurence,  and  dividing  this  sum  by  the  total 
number  of  measures.  The  formula  is: 

Z f .m 

N 

where  m equals  the  grade  or  score,  f the  frequency  of  that  grade 
and  N the  total  number  of  measures,  which  is  977  in  this  case.  The 
average  or  mean  for  this  table,  obtained  in  this  way,  is  So. 96. 

1 

The  Median 

The  median  or  that  point  on  the  scale  of  the  frequency  dis- 
tribution on  each  side  of  which  one-half  the  measures  falls,  is 
here  the  grade  of  the  person  whose  order  of  ran*c  is  four  hundred 
eighty-nine.  This  score  falls  between  67  and  68  and  by  interpola- 
tion is  found  to  be  67.35.  It  is  to  be  noted  that  the  median  is 
above  the  mean  which  indicates  that  the  frequency  is  greater  in 
the  lower  part  of  the  range  than  it  is  in  the  upper  part. 

2 

Q.uartile  Deviation 

The  quartile  deviation,  defined  as  one-half  the  distance 
between  the  first  and  third  quarter  points  in  the  distribution, 
equals  18.39. 

Influence  of  Class  Interval  on  the  Mean. 

Since  the  range  of  grades  is  from  11  to  1JQ,  or  a range  of 
89  units,  the  data  must  be  grouped  into  some  sets  of  intervals 

1.  Hugg  , Statistical  Methods,  p.  109. 

2.  Rugg,  Statistical  Methods,  p.  155. 


. 

. 


. 


. . 


9 


greater  than  one  for  convenience  in  statistical  work.  After  using 
various  intervals  to  group  the  data  for  a histogram,  the  interval 
of  five  units  was  chosen  as  the  best  since  it  seemed,  to  fit  this 
particular  data  "better  than  intervals  of  four,  3ix,  eight  or  ten. 
The  intervals  of  both  four  and  eight  units  give  very  irregular 
histograms.  The  interval  of  ten  units  actually  gives  the  most 
regular  histogram  but  that  interval  divides  a range  of  89  units 

into  only  nine  classes  and  it  is  usually  considered  best  to  work 

1 

with  at  least  twelve  to  fifteen  intervals.  Since  an  interval 
of  five  units  groups  the  grades  in  nineteen  classes  and  since  the 
histogram  for  the  unit  five  is  fairly  regular,  it  was  chosen  as 
an  apparently  good  interval.  A later  study  of  the  mean  and  the 
standard  deviation  for  each  of  these  several  intervals  shows  the 
interval  of  five  to  be  the  best. 

Taking  five  as  the  class  interval,  the  grao.es  arrange  them- 
selves in  the  frequency  distribution  as  shown  below.  The  mean  is 

2 

found  by  a short  method  in  which  an  assumed  mean  is  found  by 
inspection  and  a correction  made  by  adding  to  it  the  sum  of  the 
positive  and  negative  deviations  around  the  assumed  mean  divided 
by  the  sum  of  the  f r equenc ie s . This  may  be  stated  in  formula: 

M = + D - D 


1 F 


+ m 


Where  M is  the  true  mean,  m the  assumed  mean,  D the  deviations 
from  the  assumed  mean,  ana  F the  frequency  of  any  measure.  In  the 
table  below  the  mean  is  seen  by  inspection  to  lie  in  the  interval 
between  62.5  and  67.5  and  is  assumed  to  be  65,  which  is  the  mid- 
point of  the  interval. 


1 . Ib id . p . 86 . 

2.  Ibid.  p.  121. 


. 


. 


■ I 


10 

Distribution 

Table . 

Class 

Interval 

Fre  quency 

Deviation  (d) 
from 

Assumed  mean 

f x d 

7.5 

- 12.5 

2 

-11 

- 22 

12.5 

17.5 

5 

-10 

-50 

17.5 

22.5 

6 

- 9 

-54 

22.5 

27.5 

8 

-8 

-64 

27.5 

32. 5 

18 

-7 

-126 

32.5 

37.5 

38 

-6 

-228 

37.5 

42.5 

27 

-5 

-135 

42.5 

47.5 

54 

-4 

-216 

47.5 

52.5 

72 

-3 

-216 

52. 5 

57.5 

59 

-2 

-118 

57.5 

62.5 

86 

-1 

-86 

62.5 

67.5 

108 

0 

-1315 

67.5 

72.5 

97 

1 

97 

72.5 

77.5 

105 

2 

210 

77.5 

82.5 

104 

3 

312 

82.5 

87.5 

91 

4 

364 

87.5 

92.5 

60 

5 

300 

92.  5 

97.5 

32 

6 

192 

97.5 

102.  5 

5 

7 

35 

977 

1510 
- 1315 
195 

Assumed  mean  a 

65.  Correction 

195 
= 97? 

= .995 

65  t 

.995  = 66- 

995  s true  mean. 

11. 


When  the  grades  were  grouped  by  intervals  or  six,  eight,  ana 
ten,  the  mean  was  found  in  each  case  and  the  results  are  here  set 
down  in  tabular  form. 


Interval 

Mean 

Variation  from 
true  mean 

1 

65 . 960 

.000 

5 

65.995 

.035 

6 

66.195 

.235 

8 

66.617 

.657 

10 

66 . 390 

.430 

The  mean  obtaineu  when  the  class  interval  is  five  units 
varies  less  from  the  true  mean  than  uo  the  other  three. 

1 

Standard  Deviation 

In  verifying  the  choice  of  the  class  interval  the  standard 
deviation  was  next  considered.  The  standard  deviation  cT',  is  a 
unit  measure  or  variability.  If  on  a probability  curve,  a 
distance  equal  to  the  standard  deviation  be  laid  off  on  the  base 
line  on  each  side  of  the  mean,  and  if  ordinates  be  erected  from 
these  points  on  the  base  line  and  extended  to  cut  the  curve,  then 
Between  the  base  line,  the  ordinates  and  the  curve,  there  will 
be  included  in  this  area  68.26  percent  oi  the  measures  represented 
by  the  total  area.  The  standard  deviation  is  given  by  the 
f ormula : 

when  f is  the  frequency  of  occurence  of  any  class  interval,  d is 
the  deviation  of  the  interval  from  the  mean,  and  w is  the  total 
number  of  measures. 


1.  Ibid.  pp. 167-173 


. 

• 

■ 

< 


12 

The  work  is  much  shortened  oy  assuming  a mean  at  the  midpoint  of 
some  clas3  interval  instead  oi'  finding  the  actual  deviations  about 
the  true  mean  which  is  a number  involving  three  decimal  places.  Then 
if  the  deviations  are  laid  off  in  units  oi  class  intervals  instead 
of  one,  the  arithmetic  work  is  reduced  to  a minimum.  The  table 
showing  the  standard  deviation  for  the  interval  five  is  shown  at 
this  point  but  only  the  results  are  stated  for  the  intervals  one, 
six,  eight,  and  ten. 


Computation  of  Standard  Deviation. 


Interval 

Frequency  (f)  Deviation 

JAl  _ * 

f3T 

fd 

7.5-12. 5 

2 

-li 

121 

242 

-22 

12.5-17.5 

5 

-10 

loo 

500 

-50 

17.5-22.5 

6 

-9 

81 

486 

-54 

22. 5-27.5 

8 

-8 

64 

512 

-64 

27.5-32.5 

18 

-7 

49 

882 

-126 

32.5-37.5 

38 

-6 

36 

1368 

-228 

37.5-42.5 

27 

-5 

25 

675 

-135 

42.5-47 . 5 

54 

-4 

16 

864 

-216 

^ • O • <2  • D 

72 

-3 

9 

648 

-216 

52.5-57.5 

59 

— C, 

4 

236 

-118 

57.6-62.5 

86 

-1 

1 

86 

-86 

62. 5-67.5 

108 

0 

0 

0 

-1315 

67.5-72.5 

97 

1 

1 

97 

97 

72.5-77.5 

105 

2 

4 

420 

210 

77.5-82.5 

104 

3 

9 

936 

312 

82.5-87.5 

91 

4 

16 

1456 

364 

87.5-92.5 

60 

5 

25 

1500 

300 

92.5-97.5 

32 

6 

36 

1152 

192 

97.5-102.5 

5 

7 

49 

245 

35 

(T  - 

C o-l/vjl 

nx 

u 

<r  - 

V 0 — c'/vx  '"vjl 

- (-YTy)^  = -03^ 

- ' s 305'  _ / z.siy 

f 7 7 

r=  3.^0 i 

/X^Oi>  / 5~/0 

-/ 

/ > ^ 

<r~ 

3,  s y 

' /7.(oO 

13 


The  standard  deviation  for  the  various  intervals  ana  the 
variation  from  the  true  is  noted  here. 

Interval  Standard  Deviation  Variation  from  true  (T 

1 17.57  .00 

5 17.60  .03 

6 17.70  .13 

8 17.73  .16 

10  17.94  .37 


Since  for  the  interval  five  the  variation  of  the  mean  and 
the  standard  deviation  from  the  true  mean  ana  true  deviation  is 
less  than  for  any  of  the  other  intervals  considered,  and  since 
this  variation  is  so  small  in  both  cases,  and  the  histogram  for 
this  interval  is  fairly  regular,  this  five  unit  interval  will  be 
used  in  arranging  the  data  for  the  probability  curve. 

The  Histogram  and  Probability  Curve . 

In  the  histogram  the  class  intervals  are  laid  off  on  the  base 
line.  The  area  of  the  rectangles  is  laid  off  to  represent  the 
number  of  measures  falling  at  each  interval.  Smoothing  the  histo- 
gram by  a rough  method  of  averaging,  gives  a smoothed  curve  for 
the  data. 

The  probability  curve  was  worked  out  for  thi3  data  with  the 

aid  of  Shephard's  table.  Marking  off  on  the  x-axis  units  equal 

to  O'  on  each  side  of  the  mean,  the  value  of  y for  each  x value  is 

1 

given  by  the  formula  , 71 

y = -V  • 2 


1.  West,  Mathematical  Statistics,  p.  60 


• : 


. 


' ; 


■ 


14 


where  n is  the  total  frequency,  c7~is  the  standard,  deviation  and 

2 = ri  -rf- ' ■'  -*;h 


The  values  of  corresponding  to  the  different  values  of-^may 
be  found  by  interpolation  from  Shephard's  Table  . The  values  used 
in  plotting  are  here  shown. 

Values  for  Probability  Curve. 


X 

'cr 

2 

y 

0 

.399 

22.1445 

Cf  ~ 17.  Lo 

£ 

.352 

19 . 5360 

Y ~ - * - ^7  > 

cr  ■ • 4 

1 

. 242 

13.4610 

„ s~^  n -x 

H 

. .1505 

7 . 2430 

H 

.087 

4.8285 

2 

.054 

2.997 

z\ 

.032 

1.776 

si 

.018 

.999 

2f 

.0095 

.527 

3 

.004 

. 222 

si 

.002 

.111 

The  values  corresponding  to^rwere  found  in  Shephard's 
Table.  The  column  oi'  y values  was  obtained  by  multiplying  the 
values  by  55.511.  Plotting  these  values  the  probability  curve 
results.  The  accompanying  graph  shows  the  histogram,  with  the 
curve  obtained  by  smoothing  it,  and  also  the  true  probability  curve. 
The  curve  which  is  founa  by  smoothing  the  histogram  is  shewed 
slightly.  The  three  curves  shown  are  for  the  interval  of  five. 


1 . Ib id . p . 60  . 


16 


It  has  teen  found  that  many  traits  for  a sufficiently  large 
unselected  group  arrange  themselves  in  a curve  which  we  call  the 
normal  curve.  Grades  made  in  the  upper  classes  in  high  schools 
and  in  college  classes  usually  do  not  distribute  themselves  in 
this  fashion,  for  such  groups  are  selected,  groups  and  the  graaes 
are  more  frequent  along  the  upper  half  of  the  range.  It  is  un- 
usual to  find  the  graaes  in  a college  class  distributing  them- 
selves so  nearly  along  the  normal  curve  as  they  do  here. 

Types  of  Problems  Missed . 

The  types  of  problems  on  which  the  most  students  failed  can 
be  easily  read  off  from  the  next  table.  For  each  problem  the 
number  of  papers  having  zero,  one,  two,  three  or  four  is  stated. 

Problem. .Frequency. .Frequency. .Frequency. .Frequency. .Frequency 
of  zeros  of  ones  of  twos  of  threes  of  fours 


I 

41 

3 

II 

229 

23 

III 

9 

4 

IV 

32 

55 

V 

285 

7 

VI 

347 

1 

VII 

56 

82 

VIII 

119 

91 

IX 

397 

77 

X 

294 

82 

XI 

52 

4 

XII 

294 

107 

XIII 

132 

188 

35 

38 

890 

8 

8 

709 

12 

80 

872 

105 

62 

7 23 

46 

111 

528 

7 

2 

6 20 

3 

8 

828 

280 

198 

289 

67 

66 

370 

67 

67 

467 

180 

82 

659 

162 

152 

262 

36 

68 

553 

... 


Table  cont 1 d . 


17 


Problem. . 

Frequency . 
of  zeros 

.Frequency. .Frequency, 
of  ones  of  twos 

.Frequency, 
of  threes 

. Frequency 
of  fours 

XIV 

36 

13 

85 

49 

794 

XV 

181 

278 

240 

202 

76 

XVI 

362 

158 

95 

80 

282 

XVII 

315 

65 

170 

139 

288 

XVIII 

376 

147 

76 

79 

299 

XIX 

194 

27 

85 

229 

442 

XX 

452 

97 

87 

44 

297 

XXI 

226 

12 

15 

6 

718 

XXII 

343 

12 

29 

20 

573 

XXIII 

156 

25 

41 

97 

658 

XXIV 

137 

30 

131 

347 

332 

XXV 

193 

21 

515 

70 

178 

Totals 

o258 

1609 

2577 

2274 

12707 

In  noting  the 

totality  of  zeros,  fours  ana  the 

grades  in 

o e twe en , 

it  is  seen 

that  21.52  percent  of 

all  the  problems  were 

graded  zero,  6.58  percent 

graded  one , lo . 

55  percent 

graded  two , 

9.31  percent  graded 

three 

and  52.0 2 percent  graded  four  or 

absolutely  correct. 

The  questions  which 

were  ausolutely  missed  by 

over  three  hundred 

people 

were  those  numbered  VI,  IX 

, XVI,  XVII, 

XVIII,  XX 

and  XXII. 

These 

are  with  the  exception  of 

VI  and  XxII 

problems 

involving 

fractions  or  radicals. 

By  comparing  the 

failure  s 

as  listed 

in  the 

preceding  table 

with  the  questions,  one 

sees  that 

the  other 

questions  on  fractions  and  radicals  all  had  a 

large  number  of  failures. 


i.  . 


■ 

. 


. 


. 

. 


. 


18 

One  sees  by  reference  to  the  questions  that  a large  number  of 
the  questions,  ten  in  fact,  involve  fractions,  radicals,  or 
fractional  exponents.  The  questions  no  doubt  put  more  emphasis 
on  that  type  of  question  than  a committee  of  high  school  teachers 
would.  Many  high  school  teachers  would  consider  this  list  of 
questions  too  difficult  as  a test  of  a student’s  knowledge  of 
high  school  work.  But  as  a test  of  ability  or  a test  for  measure- 
ment purposes  it  is  good  because  the  results  do  snow  accurately 
the  wide  range  of  ability  amoung  the  students  and  allow  of  a 
possible  reorganization  of  the  classes  which  a shorter,  easier 
test  would  not  do.  The  test  was  probably  not  too  long  because 
the  majority  of  students  worked  on  all  the  questions,  and  there 
was  no  large  decrease  in  the  number  solving  correctly  the  last 
part  of  the  list. 

Cone lus ions . 

The  results  of  the  first  test  show  a great  range  in  student 
ability  in  Freshman  Algebra.  The  fact  that  one-thiru  of  the 
students  scored  less  than  60  in  such  an  examination  following  a 
review,  shows  a need  for  some  reorganization.  Whether  this 
should  be  by  means  of  a redistribution  of  the  students  in  classes, 
such  as  zero  sections,  star  sections,  and  average  sections,  or  by 
a method  of  entire  elimination  from  the  course  is  left  for 
discussion  in  the  chapter  on  correlation. 


, 


. 


IS 


Chapter  II. 

The__Final  Examination . 

The  semester  examination  was  conducted  in  essentially  the 
same  manner  as  the  preliminary  examination.  The  questions,  fewer 
in  number  but  more  complex  in  character,  are  given  below. 

Mathematics  2. 

Answer  any  ten  questions. 

Time,  three  hours. 

1.  Simplify  1 - x^ 

x 

1 + - 

1 +■  x 2x 

i-x~ 


Solve 

f or 

y the  fol 

low 

ing 

sys 

tern 

by  determinant 

s 

( 

x - 

7 - 

z 

-6 

i 

2x  +■ 

y + 

z 

— 

0 

( 

3x  - 

5y-*-8 

z 

= 

13 

Find 

t 

he 

solut 

ions 

of 

2 

X 

- bx 

+ 2 

x^  - 5x  1 10 

— 

-2 

Date 

rm 

ine 

K so 

that 

the  e 

quat 

ion 

xJ+-  4x  - 2k  = 

0, 

sat 

i s f i 8 s 

the 

gi 

van 

condi t i on 

in 

each  c 

ase  : 

: (a)  one  root 

is 

i; 

w 

the 

tw 

o r 

00  t s 

are  e 

qual ; 

(c> 

the 

product  of  the 

r 

oot  s 

is  3 . 

5.  Solve  for  x and  y 


x - 2xy+  b = 0 

(x  -y)~  - 4 - 0 


6.  Obtain  the  1st,  2nd,  3rd  and  ?th  terms  in  the  expansion  of 

( ^r-  y)8. 


7.  The  second  term  of  a geometric  progression  is  4 and  the  fifth 
term  is  -32.  Find  the  first  term  and  the  sum  of  the  first 
seven  terms. 


8.  Find  all  the  roots  of  x 


3 .2 

- x -3x  -f-  4x  - 4 — 0 


. 


. 


. 


• . 


20 


9.  Given  log  2 = 0.3010,  log  3 = 0.4771.  Find 


10.  Answer  one  or  the  following:  (a)  From  11  men  how  many 

committees  of  4 men  can  be  selected,  when  one  man  is  always 
included  on  the  committee?  (b)  If  2 balls  are  drawn  from 

a bag  containing  3 black  ana  5 white  balls,  what  is  the 
chance  that  they  will  both  be  black? 

11.  Answer  one  of  tfce  following:  (a)  Derive  the  formula  for 

the  roots  of  ax6f  bx  t c = 0.  (b)  Derive  the  formula 

for  the  sum  of  n terms  in  arithmetic  progression. 

12.  A rectangular  lawn  is  9o  feet  long  and  60  feet  wide. 

How  wide  a strip  must  be  cut  around  it  when  mowing  the  grass 
to  have  cut  half  of  it? 


Distribution  of  Grades . 


The  number  of  students  who  took  the  semester  examination 
was  nine  hundred  eight,  or  sixty  nine  less  than  took  the  October 
examination.  The  grades  for  the  nine  hundred  eight  stuaent3  are 
arranged  in  the  following  frequency  table. 


Grade 

Frequency 

Grade 

Fre  quency 

Grade 

Frequency 

100 

11 

90 

20 

80 

21 

99 

5 

89 

16 

79 

10 

98 

15 

88 

17 

78 

14 

97 

14 

87 

17 

77 

12 

96 

15 

86 

19 

76 

9 

95 

11 

85 

18 

75 

15 

94 

2 

84 

15 

74 

19 

93 

17 

83 

15 

73 

14 

92 

10 

82 

13 

72 

17 

91 

10 

81 

7 

71 

12 

. 


. 


* 


. 

. 


. 


21 


Frequency  taole  cont'd. 


Grade 

Frecjuency 

Grade 

Frequency 

Grade 

Frequency 

?0 

16 

46 

9 

22 

0 

69 

14 

45 

11 

21 

4 

68 

12 

44 

6 

20 

3 

67 

21 

43 

12 

19 

0 

66 

17 

42 

11 

18 

3 

65 

21 

41 

9 

17 

2 

64 

19 

40 

7 

16 

3 

63 

12 

39 

5 

15 

1 

62 

11 

38 

5 

14 

1 

61 

14 

37 

2 

13 

3 

60 

20 

36 

6 

12 

2 

59 

10 

35 

6 

11 

2 

58 

14 

34 

5 

10 

4 

57 

16 

33 

3 

9 

4 

56 

12 

32 

7 

8 

2 

55 

11 

31 

1 

7 

0 

54 

12 

30 

2 

6 

3 

53 

8 

29 

3 

5 

1 

52 

13 

28 

5 

4 

1 

51 

10 

27 

3 

3 

3 

50 

7 

26 

1 

2 

2 

49 

9 

25 

3 

1 

2 

43 

6 

24 

2 

0 

13 

47 

8 

23 

2 

Total 

908 

' 


22 


It  i s to  be  noted  that  there  is  an  increase  at  the 
extremities  of  the  range  in  this  table  as  compared  with  the  other 
examination.  In  the  first  examination  the  lowest  graae  was  11, 
and  only  nine  graa.es  fell  below  20.  Likewise  the  graaes  over 
90  were  few,  seventy  in  fact.  hut  in  the  preceding  table,  show- 
ing the  distribution  of  grades  for  the  second  examination,  there 
are  thirty-nine  grades  below  12  and  fifty-two  below  20,  but  on 
the  other  hand  there  is  an  increase  near  the  upper  limit,  for 
one  hundred  thirty  students  have  grades  of  SO  or  above.  It  is 
interesting  to  discover  just  how  the  change  in  distrioution 
affects  the  mean,  median  and  standard  deviation. 


Mean,  Median  and  Standard  Deviation . 

Since  the  total  frequency  is  nine  hundred  eight,  the 
median  lies  between  the  four  hundred  fifty  fourth  and  four 
hundred  fifty  fifth  score  when  they  are  arranged  in  order  of 
ranx.  by  interpolat ion  we  find: 

Median  «■  67.881 

The  calculation  used  for  computing  the  mean  and  the  standard 
deviation  follows.  The  results  obtained  are: 

Mean  **  64.879 

Standard  Deviation  — 23.45 


. 


23 


2 


Ulase  Interval . . .Frequency. .Deviation. 

. .fd. . 

. .fd 

100  - 9?. 5 

31 

7 

217 

1519 

97 . 5--92 . 5 

59 

6 

354 

2124 

92.5-  87.5 

73 

5 

365 

1825 

87.5-  82.5 

84 

4 

336 

1344 

82.5-  77.5 

65 

3 

195 

585 

77.5-  72.5 

69 

2 

138 

276 

72.5-  67.5 

71 

1 

71 

71 

67.5-  62.5 

90 

0 

1676 

62.5-  57.5 

69 

-1 

-69 

69 

57.5-  52.5 

59 

-2 

-118 

236 

52.5-  47.5 

45 

-3 

-135 

405 

47.5-  42.5 

46 

-4 

-184 

736 

42.5-  37.5 

37 

-5 

-185 

925 

37.5-  32.5 

22 

-6 

-132 

79  2 

32.5-  27.5 

18 

-7 

-126 

882 

27.5-  22.5 

11 

-8 

-88 

704 

22.5-  17.5 

10 

-9 

-90 

810 

17.5-  12.5 

10 

-10 

-100 

1000 

12.5-  7.5 

14 

-11 

-154 

1694 

7.5-  2.5 

8 

-12 

-96 

1152 

2.5-  0 

17 

-13 

-221 

2873 

908 

-1698 

20022 

1676 

-22 

: 1AAJ?  Ccx  V '■  & 

V V 

u. 

— (o  ^ 

(T*  ^ 

- ©a; 

I*  r <2  00^2. 
<jD  T 

5a.M'-,wy 

C 7C)Zj  ~ 

^ £ -2 

. oyyr 

CT*  ~ 

yvr  x ^ 

- 

b 7 * *T 

- 

t-3  Tn.-»--  wmir 


24 

The  median  is  slightly  higher  than  for  the  preliminary  examination 

in  which  we  found  it  to  be  67.35.  The  mean  or  average  is  lower, 

64.879  as  compared  with  the  65.995  of  the  previous  examination.  The 

mean  is  always  affected  by  great  variations  at  the  extremities  of 

the  range  s.na  in  this  case  the  increasing  frequency  near  the  lower 

limit  of  the  range  affects  the  mean  more  than  the  increase  near 

the  upper  limit,  because  the  mean  lies  at  a greater  distance  from 

the  lower  limit.  The  standard  deviation  for  this  idis tr ihution  is 

23.45  whereas  before  it  was  17.60.  The  magnitude  of  : tells 

1 

considerable  about  the  distribution.  The  difference  in  the 
means  of  the  two  distributions  is  about  one  unit,  yet  the  devia- 
tion around  the  mean  is  much  greater  in  this  second  distribution 
than  in  the  first.  A large  value  of  the  standard  deviation  is 
due  to  a relatively  high  frequency  of  the  variates  near  the  extremes 
of  the  di s t r ibut ion . 

P rob  ability  Curve  . 

Prom  the  9tudy  of  the  range,  the  distribution  and  the  standard 
deviation  it  is  seen  in  advance  that  the  probability  curve  will  be 
less  steep,  ana  broader  at  the  base  than  the  curve  for  the  data  from 
the  preliminary  examination.  We  find  as  before  that  the  smoothed 
curve  from  the  histogram,  though  slightly  skewed,  follows  the 
general  shape  of  the  probability  curve  much  more  closely  than  is 
expected  from  any  college  group.  Because  of  the  elimination  taking 
place  throughout  grade  and  high  school  work,  the  grades  of  college 
Freshmen  seldom  conform  so  closely  to  a probability  curve. 


. 


- 


26 


Pj-.BCussion  of  Semester  Grades . 

The  distribution  of  semester  grades  in  Mathematics  2,  College 
Algebra,  is  clearly  shown  in  the  following  table,  where  the 
frequency  of  each  literal  grade  and  its  percentage  of  the  total 
frequency  are  both  stated.  A,  B,  C ana  D are  passing  grades;  E 
is  failure;  W is  withdrawal,  and  hr.  means  dropped. 


Grade 

frequency 

Percentage  of 
total  frequency 
through  semester 

Percentage  of 
total  frequency 
enrolled 

A 

111 

12.09 

10.22 

3 

186 

20.26 

17.12 

C 

211 

22.98 

19.43 

D 

208 

22 . 65 

19.15 

E 

202 

22.00 

18.60 

1 

Total 

918 

99.88 

Dr.  or  W. 

168 

15.47 

Total 

1086 

99.99 

The  percentages  based  on  the  total  enrollment  do  not  give  a 
fair  basis  for  discussion  since  most  of  the  withdrawals  occurred 
by  change  of  study  list  in  the  first  week  after  registration. 


1.  Semester  averages  were  available  for  ten  students  who  took 
the  final  examination  late  and  whose  grades  are  not  included 
in  the  frequency  table  of  final  examination  grades. 


, 


. 


. 


- 


27 


About  one  hunured  of  the  one  hundred  sixty-eight  withdrawals 
occurred  before  the  preliminary  examination  in  October.  The 
number  of  failures  based  on  enrollment  remaining  throughout  the 
semester  is  exceedingly  high,  for  it  is  twenty-two  percent.  how- 
ever, aoout  the  same  percentage  occurs  each  first  semester  in 
1 

Mathematics  2 . It  is  frequently  said  that  there  is  a larger  per- 
centage of  failures  in  Mathematics  2 than  in  any  other  course  in 
the  University.  But  one  finds  single  courses  in  a number  of 
departments,  in  Zoology,  in  Accountancy,  in  Athletic  Coaching  and 
in  Economics,  where  the  failures  range  from  nineteen  to  twenty-two 
and  one  half  percent.  Courses  in  Chemistry,  Rhetoric  and  German 

run  surprisingly  high  in  the  numoer  of  E's  given,  although  they 

1 

do  not  run  as  high  as  the  ones  mentioned  aoove  . 

The  high  percentage  of  failures  in  mathematics  has  been 
interpreted  in  a number  of  ways.  It  has  caused  many  persons  to 
criticize  the  subject  matter  of  the  courses  and  to  suggest  a large 
amount  of  reorganization  of  the  material  included  in  such  courses. 
It  has  occasioned  investigation  of  the  methoas  of  teaching,  by 
educators  who  are  convinced  that  therein  lies  the  inefficiency.  It 
may  be  that  by  their  very  nature  mathematics  courses  allow  of 
more  definite  grading  and  more  frequent  checking  throughout  the 
semester,  so  that  the  final  grade  is  more  largely  a matter  of 
record  than  of  estimation  and  is  thus  apt  to  be  lower  than  a grade 
given  in  a subject  not  allowing  such  definite  records.  Careful 
investigation  and  worx  in  any  of  these  directions  will  no  doubt 


1.  Registrar's  Statistical  Report  on  the  distribution  of 
grades  in  various  courses. 


. 


. 


. 


' 


28 

do  much  to  reduce  the  number  oi'  failures,  but  as  long  as  the 

group  requirements  of  the  several  colleges  of  the  University  are 

such  that  a large  proportion  of  the  Freshmen  must  take  college 

algebra,  the  percentage  of  failures  will  continue  to  be  high. 

Other  studies  show  that  a large  number  of  failures  in  Freshman 

Mathematics  in  both  high  school  and  college  is  very  general.  A 

typical  example  is  that  given  in  a study  of  "Student  Achievements 

1 

in  Mathematics  in  Shortridge  High  School,  Indianapolis".  There 
the  failures  in  Freshman  ana  Sophomore  Mathematics  range  from 
sixteen  to  twenty  eight  percent  of  the  pupils  remaining  in  the 
course  throughout  the  semester,  or  from  twenty-four  to  thir  ty- 
four  percent  of  the  total  enrollment  in  the  courses.  Probably 
the  most  effective  method  of  reorganization  is  to  reorganize  the 
classes  after  an  early  preliminary  examination  and  to  take  care 
of  the  poorer  students  in  some  special  way.  This  problem  is 
discussed  again  in  Chapter  III. 

The  Influence  of  the  Final  Examination  on  the  Semester  Uraue. 
In  the  Mathematics  Department  the  final  examination  is 
supposed  to  count  one-third  in  determining  the  semester  average. 
To  investigate  the  weight  which  instructors  really  gave  to  the 
examination  graaes  in  estimating  the  semester  average,  the  final 
grades  which  are  below  7Q  with  the  corresponding  literal  averages 
are  listed. 

1.  E.  C.  Dodson,  School  Science  and  Mathematics,  Vol . 14. 

May,  1914. 


. 


■ 


29 


Partial  Table 

of  Jj'inal  Grades  and  Semester 

Averages . 

frequency 

Examination  Grade 

Semester  Average 

65 

23 

5 

1 

60-70 

50-60 

40-50 

21 

C 

C 

C 

C 

68 

60-70 

D 

49 

50-60 

0 

33 

40-50 

D 

8 

30-40 

D 

6 

25-30 

D 

1 

0 

D 

That  the  examination  grades  were  given  a weight  of  one-third 
in  most  cases  is  entirely  probable  from  this  tabulation.  How- 
ever, the  table  shows  some  glaring  exceptions.  A grade  of  C could 
not  be  given  when  the  final  score  was  21,  if  the  latter  counted 
one-third,  even  if  the  class  room  work  were  of  an  A standard.  Nor 
could  the  cases  in  which  the  D average  was  given  with  final  grades 
between  25  and  30  have  been  so  determined  if  the  examination  score 
counted  one-third.  More  striking  still  is  the  case  where  a student 
received  zero  in  the  examination  and  yet  passed  the  course  with 
a D average.  But  these  eight  cases  are  exceptional  and  in  each 
case  there  may  have  existed  some  special  reason  or  extenuating 
c ircumB t&nce s that  justified  the  seeming  discrepancy  in  the  two 
grades . 


. . 


* 


30 


Chapter  III. 

Correlation  of  Grades  of  the  Two  Examinations. 

Correlation  Defined. 

It  is  of  interest  in  this  investigation  to  discover  the 

relation  "between  the  results  of  the  preliminary  examination  over 

high  school  algebra  ana  the  final  examination  over  college  algebra. 

If  a correlation  between  the  two,  or  a dependence  of  one  on  the 

other  can  be  established,  then  various  uses  of  such  a preliminary 

test  can  be  determined.  Two  characteristics  are  said  to  be 

correlated  when  there  is  a tendency  for  the  changes  in  the  value 

1 

of  one  to  depend  on  the  changes  in  value  of  the  other  . 

Correlation  Table. 

In  order  to  determine  the  correlation  the  data  must  be 
arranged  in  a correlation  table.  The  distribution  is  so  arranged 
that  the  grades  of  the  preliminary  test  lie  along  the  vertical 
axis  and  those  of  the  second  along  the  horizontal  axis.  The 
correlation  table  is  shown  on  a later  page.  Certain  additional 
ro7;s  and  columns  are  shown  on  the  two  succeeding  pages  because 
of  the  size  of  the  complete  table.  The  data  for  both  examinations 
for  the  same  stuaent  wa3  available  in  eight  hundred  sixty-seven 
cases  and  the  correlation  in  this  chapter  was  based  on  that 
number,  certainly  large  enough  to  be  very  representative. 


1.  West,  Introduction  to  Mathematical  Statistics,  p.  73. 


. 


31 


In  the  table  each  student  is  located  by  his  grade  in  each  examina- 
tion. For  instance  the  number  two  in  the  upper  right  hand  corner 
indicates  that  two  students  scored  between  97.5  and  100  in  the 
first  examination  and  between  97.5  and  100  in  the  second.  The 
frequency  nine  just  below  and  in  the  same  column  indicates  that 
nine  students  scored  between  92.5  and  97.5  in  the  first  and 
between  97.5  and  100  in  the  second  examination. 


Coefficient  of  Correlation . 

Correlation  is  frequently  expressed  by  the  coefficient  of 

correlation,  T , where  V is  expressed  by  the  product  moment 
1 

formula: 

T.  _ n ■ • > t t. 

x'y'  is  obtained  by  multiplying  the  frequency  of  each 

class  by  its  deviation  from  the  horizontal  axis  and  then  by  its 
deviation  from  the  vertical  axis,  and  finding  the  sum  of  these 
products.  C and  Cv  are  the  corrections  to  be  made  about  the 
assumed  means  in  the  x-array  and  the  y-array,  respectively.  The 
horizontal  axis  is  drawn  through  the  assumed  mean  of  the  y-array 
and  the  vertical  axis  through  the  assumed  mean  of  the  x-array. 

C Tjt  and  <7y  are  the  standard  deviations  of  the  two  arrays. 
These  terms  C , C , have  been  used  in  the  calculation  of 

x y 

the  mean  and  of  the  standard  deviation.  P.  E.  is  the  probable 
error,  to  be  defined  in  the  next  topic.  The  ar ithrae t ical  work 
involved  in  the  calculation  of  r is  given  in  detail  with  the 
correlation  table  and  the  resulting  value  i3:  r=  .673  p.E. 


1.  Rugg  , Statistical  Methods  Applied  to  Education,  p.  269 


. 


. 


. 


* 


. 


32. 


Probable  Error 

The  probable  error  in  any  result  may  be  defined  as  that 

deviation  from  the  determined  value  such  that  it  is  an  even  wager 

that  the  true  value  lies  within  this  amount  of  the  determined  value 
2 

The  formula  for  the  probable  error  in  the  case  of  the  correlation 
coefficient  is: 

P.  E . of  r = .67449  (l-r*S 

i~ir 

where  r = coefficient  of  correlation, 
w = frequency  of  distribution. 

Eor  this  data 

p £.  m .£  7Y?7 ( / - 'L  - .0  13 

Ou+4  T -.i.  73  31.0  13 

3 

The  Regression  Lines. 

The  degree  of  correlation  in  the  data  is  indicated  by  the 
variation  or  the  means  from  array  to  array.  The  variation  in  the 
means  of  the  arrays  is  shown  graphically  by  the  curve  of  the  means 
which  is  called  a regression  curve.  There  are  two  regression 
curves,  one  corresponding  to  each  set  of  arrays.  A straight 
line  fitted  to  the  means  of  the  arrays  is  called  a line  of  re- 
gression. If  the  regression  curves  approximate  straight  lines, 
the  regression  is  said  to  be  linear.  The  slope  of  the  regression 
lines  depends  on  the  standard  deviations  of  the  two  arrays.  The 
equations  of  the  lines  of  regression  are: 


T 


fy 


1.  Rietz  and  Shade.  U.  of  J. 

2.  Rugg , Statistical  Methods 

3.  West,  Introduction  to  Mathematical  Statistics,  p.  ^4. 


Studies 

applied 


Vol.  Ill  p.  17. 
to  Education,  p.  272. 


' 


. 

- • 

- 


. 


33 


In  this  correlation 
and  by  substituting 
become 


the  values  or  r,  C^-and  1 have  been  determined 
these  values  in  the  auove  equations,  they 


- JC 


Jr 


which  are  the  equations  of  the  two  lines  of  relationship. 

1 

Correlation  Ratios . 

If  the  means  of  the  correlation  table  do  not  accord  fairly 
well  with  a straight  line,  the  coerficient  r and  the  regression 
equations  will  not  describe  the  relationship  of  the  two  traits 
unaer  consideration,  for  the  coefficient  depends  upon  the  fact 
that  the  means  of  the  arrays  lie  not  far  from  the  line  of 
regression.  If  the  means  of  the  table  do  not  fall  approximately 
on  a straight  line,  the  relationship  is  expressed  by  the  correla- 
tion ratio  >[,  which  is  the  ratio  of  the  standard  deviation  of 
the  arithmetic  means  of  each  of  the  columns  (or  rows)  of  the 
table  to  the  standard  deviation  of  the  whole  table  itself.  There 
are  two  values  of  one  for  each  array.  expresses  the 

dependence  of  y upon  x,  or  in  this  table  the  dependence  of  the 
preliminary  examination  upon  the  final;  ^expresses  the  dependence 
of  x upon  y,  or  of  the  final  examination  upon  the  preliminary. 

The  latter  ratio  is  more  significant  in  this  study  because  if  the 
second  examination  depends  very  strongly  on  the  first,  the 
preliminary  examination  will  have  a prognostic  value. 


1.  Rugg,  Statistical  Methods,  p.  276-282. 
1.  West,  Mathematical  Statistics,  p.  76. 


■V 


. 


. 


. . 

. 


34 


To  calculate  the  values,  aaa  two  rows  and  two  columns  to  the 
correlation  taole.  One  the  ( JE?  X 0 column  is  merely  the  square  of 
the  XXao  lumn  already  in  the  table.  The  second  is  obtained  by 
dividing  this  £ -X^column  by  the  frequencies  or  the  y arrays. 

The  formulae  for  the  correlation  ratios  are: 


y 


Vr 


(7 


■3T" 


y 


The  calculation  is  included  in  the  correlation  table.  The  values 
obtained  are: 

= .6766  yl  = .6372 


K 


Blakeman  Test  . 

In  order  to  determine  whether  or  not  the  correlation  table 
exhibits  linear  regression,  that  is,  whether  we  can  use  the  pro- 
duct moment  formula,  we  use  the  Blakeman  criterion  for  linearity. 
2 

It  is  that  if 

J\T  - r V <21 

then  the  distribution  shows  linear  regression.  The  test  in  this 
correlation  shows  that  the  regression  is  non  linear  for  the 
value,  or  the  dependence  of  the  first  examination  on  the  second. 

M -.r*)  - Y, 

Jf  - r’)  - /<*■  >U 


1.  Bugg , Statistical  Methods,  p.  283. 

2.  Professor  A.  h . Grathorne  suggested  this  simplified  form 


: 


. 


■ 


35 


Since  the  regression  is  nofi  linear  the  regression  lines  cLo  not  i'it 
the  curve  through  the  means  or  the  arrays.  This  is  shown  by 
plotting  the  means  of  the  arrays  ana  noting  how  far  they  fall 
from  the  regression  lines.  The  values  to  be  plotted  are  shown 
in  the  last  column  ana  last  row  or  the  table. 


S c atter  .Diagram . 

The  correlation  table  is  given  in  much  detail.  The  worf  in 
rinaing  r,  and  , follows  rather  closely  the  methods  used 

by  Hugg  ana  by  West  except  for  some  s impl ir ica r ion  which  was 
suggested  oy  Professor  Urathorne.  The  correlation  ratios  can 
be  obtained  directly  from  the  table,  oy  the  addition  or  columns 
7 and  8,  and  the  corresponding  rows.  The  results  of  the  work 
have  been  stated  in  the  preceeaing  topics  hut  they  can  be  more 
readily  understood  when  the  correlation  table  is  put  in  the  form 
of  a scatter  diagram.  Such  a diagram  is  shown  in  the  next  table. 
The  numbers  representing  the  frequency  in  each  interval  are 
replaced  by  the  correct  number  of  dots.  The  diagram  contains 
eight  hundred  sixty  seven  dots  each  one  located  correctly  with 
reference  to  its  position  in  the  two  examinations.  The  axes 
are  two  lines  drawn  tnrough  the  means  of  the  x ana  y arrays,. 
Taking  the  intersection  of  these  two  lines  as  the  center  or 
origin,  the  regression  lines  are  plotted  from  the  equations 
previously  formulated.  The  means  of  the  arrays  through  which 
the  regression  curves  should  pass  are  plotted  from  the  values 
in  column  9 and  row  9,  respectively.  In  the  diagram  the 
positive  direction  is  to  the  right  and  upwards  and  the  negative 
to  the  left  and  downwards. 


' 


. 


. 


36 


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40 


If  each  student  made  exactly  the  same  grade  on  each  examina- 
tion, the  two  show  an  absolute  dependence,  one  on  the  other  and 
a perfect  correlation  is  said  to  exist.  In  such  a case  all  the 
frequencies  or  the  dots  in  the  scatter  diagram  would  fall  on  the 
main  diagonal,  a line  making  45  degree  angles  with  the  horizontal 
and  vertical  axes.  The  higher  the  correlation  the  more  nearly  do 
the  dots  arrange  themselves  on  the  main  diagonal.  In  a perfect 
correlation,  the  value  of  r or  >|  is  one;  if  no  correlation  exists 
the  value  of  r or  i|  is  zero.  Between  these  two  lie  all  degrees  of 
correlation. 

There  are  many  opinions  as  to  what  constitutes  a high 

correlation  as  is  evident  from  a study  of  educational  reports 

1 

involving  correlation  problems.  Rietz  found  that  the  coefficient 
of  correlation  for  stature  of  father  and  son  is  .596  and  for 
stature  of  brother  and  brother  is  .591.  We  think  of  stature  of 
father  and  son  as  closely  related  variates  so  Rietz  concludes 
that  anything  greater  than  .4  is  a high  correlation.  A reference 
to  a series  of  correlations  made  by  Professor  Crathorne  for 
students  in  Mathematics  courses  in  the  University  gives  further 
standards  for  an  interpretation  of  the  magnitude  of  correlation 
which  is  to  be  considered  significant.  A correlation  between 
grades  made  in  Freshman  Algebra  and  Trigonometry  taken  under 
the  same  instructor  and  taken  in  the  same  semester  for  two 
thousand  cases  shows  a value  of  J\J  = .?3?±.Q07 


1.  Rietz  and  Shade.  U.  of  I.  Studies.  Vol.III.  p.  19 


* 


41 


A correlation  between  Freshman  Algebra  taken  in  the  same  semester 
under  different  instructors  for  four  hundred  students  gives  a 
value  of  r = .620  J-.Q21 

A correlation  for  Mathematics  7 and  9,  two  successive  courses 
in  Calculus,  under  the  same  instructor  and  with  the  same  text, 
for  two  hundred  and  fifty  students  gives  a value 

r = . 330  ± .013 

These  three  cases  are  ones  in  which  the  relationship  is  known 
to  be  very  great  so  that  the  resulting  correlations  are  considered 
high.  Students  taking  Calculus  for  two  semesters  unaer  the  same 
instructor  would  in  most  cases  receive  identically  the  same 
grade  and  a correlation  ootained  in  such  a case  is  to  be  thought 
of  as  unusually  high.  Likewise  a correlation  obtained  for 
parallel  courses  in  Algebra  and  Trigonometry  is  thought  of  as 
high.  Since  the  correlation  between  the  preliminary  and  final 
examinations  in  this  present  discussion  is  .673  ana  lies  between 
the  correlation  of  .620  for  Algebra  and  Trigonometry  under 
different  instructors,  and  that  of  .737  for  Algebra  and 
Trigonometry  under  the  same  instructors,  it  must  be  a high 
correlation. 

C onclusions  and  Suggestions . 

A correlation  of  .673  such  as  we  have  in  this  table  is  very 
high  when  compared  with  the  correlations  cited  aoove.  In  the 
scatter  diagram  the  slope  of  the  regression  lines  shows  the 
broad  general  tendency  of  the  traits.  The  rate  at  which  one 
trait  increases  with  the  other  depends  upon  the  slope  of  its 
regression  line,  so  the  relationship  can  be  read  from  these  lines. 


’ 


. 


. 


' 


. 


. 


42 


When  the  true  means  are  used  instead  of  the  regression  means,  in 
this  diagram,  the  regression  curves  would  pass  through  the  dots 
indicated  by  crosses  for  the  one  case  and  by  circles  in  the 
second  case,  which  do  not  coincide  with  our  regression  lines,  as 
one  sees  at  once  from  the  diagram. 

In  the  consideration  of  the  prognostic  value  of  the  preliminary 
examination  it  is  essential  to  consider  all  the  data  as  set  forth 
in  the  scatter  diagram,  rather  than  to  draw  sweeping  conclusions 
from  the  high  correlation  coefficient  as  is  so  frequently  done  in 
educational  studies.  The  correlation  coefficient  or  ratios 
express  a relationship  for  the  group  as  a whole  but  they  say 
nothing  certain  regarding  the  individuals.  There  may  be  a number 
of  cases  which  show  absolutely  no  relationship  in  the  two  traits, 
out  on  inspection  of  our  correlation  ratios  will  fail  to  indicate 
this.  Inspection  of  the  diagram  shows  that  of  the  seventy-seven 
students  scoring  less  than  42.6  on  the  first  examination  only 
three  scored  more  than  67.6  on  the  other  examination.  One  hundred 
seventeen  scored  less  than  47.6  on  the  first  examination  and  of 
these  only  thirteen  scored  above  67.6  on  the  second.  From  the 
one  hundred  seventy  nine  who  were  graded  below  52.5  on  the  first 
examination,  only  twenty-four  were  above  67.5  in  the  final.  The 
indications  then  are  that  if  the  students  who  made  the  very  low 
grades,  say  below  50,  on  the  first  examination,  were  eliminated 
from  the  course  at  that  time,  or  transferred  to  some  course 
adapted  to  their  needs,  the  number  of  failures  in  the  Freshman 
Algebra  classes  would  be  reduced  by  half. 


. 


. 


. 

. 


. 


. 


43 

Another  consideration  here  adds  its  weight  to  the  argument 
for  the  prognostic  value  of  this  preliminary  test.  That  is  the 
consideration  of  the  record  of  those  students  who  withdrew  before 
the  final  examination.  hy  reference  to  the  table  of  semester  grades 
in  Chapter  II,  it  is  found  that  one  hundred  sixty-eight  students 
withdrew  before  the  final  examination.  Of  these  ninty-nine  with- 
drew even  before  the  preliminary  test,  and  so  do  not  enter  into 
this  study  in  any  way.  Of  the  sixty  nine  who  withdrew  in  the 
period  between  the  two  examinations  the  tabulation  of  results  from 
the  first  test  shows  that  in  a majority  of  cases  these  were  weam 
students  who  eliminated  themselves.  So  it  appears  that  the  stuaents 
who  did  very  poor  work  in  the  October  examination,  did  almost  as 
poor  work  in  the  final,  or  in  many  cases  eliminated  themselves 
before  the  final.  The  table  shows  the  record  of  such  students. 

First  Examination  Record  of  Stuaents  Withdrawing 

before  final  examination. 


Grade 

Frequency 

Grade 

Frequency 

90-100 

4 

40-50 

15 

80-90 

1 

30-40 

10 

70-80 

6 

20-30 

4 

60-70 

10 

10-20 

4 

50-60 

15 

0-10 

0 

Total . . . 

The  method  to  be  adopted  in  a reorganization  of  the  Freshman 
Algebra  course  following  an  examination  over  high  school  algebra, 


. 


44 


is  to  a large  extent  an  administrative  problem.  There  might  be 
merely  a shirting  or  students  within  the  classes,  so  that  the  very 
poor  ones  form  "zero”  sections,  and  the  very  superior  students  "star” 
sections.  The  Benefits  of  such  a division  are  debatable,  but  such 
an  arrangement  is  not  without  successful  precedent.  Those  who  do 
not  need  the  algebra  as  a prerequisite  for  other  courses  and  can 
secure  their  group  requirements  in  another  way  might  well  be 
eliminated  entirely  if  their  first  examination  grades  are  unusually 
low.  If  there  was  no  way  to  enter  them  in  some  other  course  at 
that  time,  such  students  could  well  oe  allowed  to  devote  their 
time  to  the  remaining  thirteen  or  fourteen  hours  which  they  are 
attempting  to  carry.  Another  problem  is  that  of  taxing  care  of 
those  students  who  are  weak  in  algebra  and  yet  must  carry  the  course 
because  it  is  a prerequisite  for  their  later  courses.  If  a student 
from  the  engineering  school  fails  in  algebra,  he  is  in  most  cases 
so  delayed,  that  his  college  course  is  prolonged  one  year.  Almost 
the  only  suggestion  here  is  that  these  students  should  be  given 
five  hours  of  algebra  a week  with  three  hours  credit.  This  method 
was  formerly  used  and  the  chief  objection  to  it  was  that  the  weaker 
students  were  required  to  do  more  work  than  the  others,  for  the 
preliminary  examination  subsequent  to  registration  meant  an 
additional  two  hours  work  on  an  already  normal  schedule.  But  since 
such  a high  correlation  is  seen  to  exist  between  a preliminary  and 
a final  examination,  there  is  nothing  to  be  said  for  allowing  the 
poor  students  to  go  on  through  the  semester  to  almost  certain 
failure,  if  it  can  be  prevented  by  some  other  method.  In  order 
that  elimination  or  reorganization  work  an  injustice  to  the  fewest 


. 


•!  . 


45 


number,  the  standard  for  elimination  should  he  low.  For  instance, 
we  have  shown  that  an  elimination  of  all  students  scoring  less 


1 

than  42.5  woulu  at  most  he  unfair  to  three.  A number  of  studies 
show  that  the  correlation  between  entrance  examinations  and 
college  work  is  much  lower  than  a correlation  between  high  school 
work  and  college  work  and  conclude  that  entrance  examinations 
are  to  he  seriously  opposed.  Professor  Lincoln  of  Harvard  states 
that  the  correlation  between  entrance  examinations  and  college 
worm  is  .*7  and  that  between  high  school  work  and  college  work 
it  is  .69.  The  preliminary  examination  of  this  discussion  was 
not  an  entrance  examination  for  it  was  given  after  eight  review 
lessons  under  college  instructors,  ana  the  coefficient  of  correla- 
tion was  .673,  much  higher  than  in  the  case  of  entrance  examina- 
tions as  cited  by  Lincoln.  This  difference  largely  removes  the 
objectionable  features  of  an  entrance  examination , but  to  be 
absolutely  fair  in  the  matter,  the  elimination  standard  should 
be  made  lower  than  the  usual  passing  mark.  A careful  considera- 
tion of  the  scatter  diagram  should  be  the  basis  of  a decision  in 
this  matter  and  in  this  case  the  conclusion  is  that  a score  of 
50  would  be  approximately  the  correct  standard  for  elimination. 


1.  Thorndike,  Educational  Review.  Vol.  XXXI. 

1 . Lincoln,  E.  A.,  Relative  Standing  in  High  School. 

Early  College  and  Entrance  Examinations. 
School  and  Society,  Vol.V. 


. ■ . 


. 


' 


. 

* I 


46 


Chapter  IV. 

Correlation  with  Respect  to  the  Time  Element. 

The  previous  chapter  on  correlation  does  not  consider  the 
element  of  time  which  is  also  a variable  for  this  data.  Many 
of  the  students  had  taken  high  school  algebra  several  years  ago, 
while  others  had  taken  such  courses  very  recently.  It  would  seem 
that  the  length  of  time  intervening  between  the  last  high  school 
course  and  this  examination  over  high  school  algeora  would  be  a 
factor  of  decided  influence.  To  accurately  determine  this  factor 
the  students  cards  were  divided  into  seven  groups,  according  to 
the  length  of  time  which  had  elapsed  since  their  high  school  work 
in  algebra.  The  stuaents  were  asked  to  give  this  data  on  their 
final  examination  books  and  only  six  hundred  eleven  did  so.  How- 
ever, this  is  a large  numoer  and  is  representative  in  its 
distribution  over  the  seven  periods  as  is  shown  in  this  table. 


Date  of  last  course 

Time 

since 

.frequency 

1921 

1 

yr  . 

ago 

74 

1920 

2 

yrs . 

ago 

172 

1919 

3 

yrs  . 

ago 

177 

1918 

4 

yrs . 

ago 

83 

1917 

5 

yrs . 

ago 

46 

1916 

6 

yrs . 

ago 

25 

1903-1915 

7- 

14  yrs,  ago 

34 

Total  611 

. 


. • 

4? 

If  the  time  element  is  non-essential  approximately  the  same 
correlation  would  be  reached  in  each  of  the  seven  groups,  when  the 
results  of  the  two  examinations  are  correlated  separately  for  each 
group.  If  the  time  element  is  noticeable,  a gradual  increase  or 
decrease  in  the  correlation  coefficient  and  ratios  will  indicate 
it.  Correlation  tables  are  given  in  each  case,  but  the  values  of 
>|  , and  ¥ together  with  the  probable  error  ana  the  Blakeman 

test  are  shown  in  the  one  table. 

Correlation  by  years. 


Since  last 
course 

r 

P.E. 

> 

blakeraan 
Test  (yjy) 

Blakeman 
Test  ()jx) 

1 year 

.7015 

.040 

.771 

7 . 57  < 1 1 

.765 

6.89  <11 

2 " 

. 703 

. 026 

.745 

10.45  < 11 

.718 

3.66  <11 

2 « 

.679 

.027 

. 692 

3.01  <11 

.728 

12.04  > 11 

4 " 

. 674 

.040 

. 804 

16.01  >11 

. 733 

6.89  <11 

5 " 

.608 

.062 

.903 

19.6  > 11 

.741 

8.25  < 11 

6 " 

.533 

.096 

.866 

11.60  > 11 

.746 

6.81 < 11 

7-14  " 

. 749 

.051 

.771 

6.46  < 11 

.765 

4.31<li 

46 


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52 


Discussion  of  rand  ')]  values  . 

In  reviewing  a large  numoer  of  eaucational  stuuies  which 
involve  statistical  methods  I have  notea  that  the  conclusions 
have  been  based  solely  on  the  coefficient  oi  correlation.  If 
only  the  values  of  T were  used  from  the  preceeding  table,  we 
could  reach  conclusions  entirely  contrary  to  those  depending  on 
a considerat ion  of  the  y values.  It  is  they!  values  which  are 
significant  for  the  hlakeman  test  shows  that  in  most  of  the  cases 
the  regression  is  non-linear,  and  that  the  correlation  coefficient 
is  not  an  accurate  measure  of  relationship.  From  the  T1  value s , 
which  show  a gradual  aecrease  with  the  increase  in  time,  one 
would  conclude  that  those  students  who  took  their  high  school 
work  some  years  ago  did  much  poorer  work  on  the  first  examination 
than  those  wno  had  taken  it  only  recently,  and  hence  were  open 
to  a greater  degree  or  improvement  during  the  semester,  so  that 
a lower  correlation  existea  between  the  two  examinations.  how- 
ever, a consideration  of  the  correlation  ratios,  which  are  the 
more  accurate  measure  of  relationship  here,  shows  no  such  gradual 
decrease  with  the  years.  Since  the  degree  of  dependence  of  the 
second  examination  upon  the  first  is  the  important  relationship, 
the  values  of  are  to  be  studied.  There  seems  to  oe  no  special 
significance  in  the  size  of  these  ratios,  although  with  the 
exception  of  1921,  the  >/x-values  increase  slightly  with  the  time, 
instead  of  decreasing.  The  high  V(  value  for  the  stuuents  in  the 
1908-15  group  is  at  first  surprising  but  if  can  he  easily  explain- 
ed in  view  of  the  fact  that  students  who  return  to  school  work 


- 


■ 


. 


■ 


53 


after  a long  period  of  time  are  no  doubt  exceptional  students  , or 
at  least  more  mature  and  more  apt  to  maintain  the  same  stanuara 
of  work  at  all  times,  leading  to  a high  correlation  between  the 
examination  graces. 

Conclusions . 

Prom  the  correlation  as  set  forth  in  this  chapter,  one  can 
draw  little  positive  conclusion  as  to  the  effect  of  the  time 
variable  in  the  investigation.  The  wor.£  in  this  chapter  is  of 
considerable  value,  however,  in  that  it  furnishes  a striking 
example  of  the  fallacy  of  establishing  the  correlation  coefficient 
and  drawing  sweeping  conclusions  therefrom,  unless  one  has  first 
studied  the  type  of  regression  and  found  that  this  coenicient 
is  a true  expression  of  relationship  for  the  problem  at  hanu.  One 
who  is  acquainted  with  the  matnemat ical  questions  involved  in  the 
interpretation  of  the  coefficient  of  correlation  is  continually 
appalled  by  the  manner  in  which  writers  of  little  mathematical 
training,  who  have  obtaineu  correlation  results  oy  biinuly 
following  a given  formula,  araw  sweeping  conclusions  from  such 
results.  With  the  present  trend  of  modern  educational  metnods 
toward  the  use  of  tests,  measurements  and  statistical  representa- 
tion, there  is  much  room  in  this  field  for  investigators  with  a 
thorough  icno'.vleuge  of  statistics  who  will  sanely  interpret  the 
results  of  their  studies. 


■ 


» 


54 


Summary. 


Certain  definite  conclusions  can  be  made  from  this  study.  In 
the  first  place,  the  results  of  the  preliminary  test  show  that  the 
students  taking  Freshman  Algebra  are  essentially  a non-selected 
group;  that  about  one-third  of  them  fall  below  60  in  such  a test 
and  that  there  is  need,  from  the  very  first  of  tne  semester  of 
some  sort  of  readjustment.  Next,  we  found  that  in  the  final 
examination  more  than  one-third  fall  uelow  60;  that  the  average 
and  median  grades  approximate  those  of  the  preliminary  test,  so 
that  there  seems  to  be  no  general  improvement  in  the  group.  In 
the  third  place,  a correlation  of  the  two  sets  of  grades  gives  a 
coefficient  equal  to  .673  which  shows  that  the  second  examination 
is  highly  dependent  on  the  preliminary  test.  A study  of  the 
scatter  diagram  shows  that  of  one  hundred  seventeen  students 
scoring  less  than  47.5  on  the  first  examination  only  thirteen  score 
above  67.5  on  the  final,  and  that  of  one  hundred  seventy-nine 
scoring  less  than  52.5  on  the  first  examination  only  twenty-four 
were  above  67.5  on  the  final.  So  if  50  were  taken  as  the  elimina- 
tion standard,  the  failures  in  Freshman  Algebra  could  be  reduced 
by  one-half.  Lastly,  a consideration  of  the  time  elapsing  since 
the  student's  last  course  in  high  school  algebra,  indicates  that 
this  is  not  a decided  factor  in  the  investigation  ana  that  with 
a review  period  previous  to  the  first  test,  this  time  element  is 
of  little  significance.  The  question  of  a reorganization  method 


. > H 


. 


. 

. 


- 


55 


is  yet  to  be  decided.  The  poorer  students  who  do  not  need  mathe- 
matics as  a prerequisite  to  other  courses,  might  well  be  eliminated 
entirely.  Those  who  must  have  it  to  enable  them  to  take  other 
courses,  should  be  put  in  a special  course  designed  to  meet  their 
needs.  This  study  clearly  indicates  that  a very  high  prognostic 
value  may  be  put  on  such  a preliminary  test  and  that  it  should 
be  given  with  such  a purpose  and  the  results  used  for  a complete 
reorganization  of  the  Freshman  work. 


' 


. 


56 


it 

.bib  1 iography . 

Beferences  to  Books . 

Rugg , Harold  0. --Statistical  Methods  Applied  to  Education. 
West,  Carl  J . Intr oauct ion  to  Mathematical  Statistics. 
Henderson,  J.  L. --Admission  to  College  by  Certificate. 

Whipple,  G.  M. --Manual  of  Mental  and  Pxiysical  Tests. 

References  to  Journals . 

Burris,  W.  P . --Correlat ion  of  Abilities  Involved  in  Secondary 

School  Work.  Columbia  University  Contributions 
to  Education  Vol.  Al.  February  1903. 

Broome,  E.  C.--A  Historical  and  Critical  Discussion  of 

College  Admission  Requirements.  Columbia 
University  Contributions  to  Education.  Vol. XI 
April  , 1903. 

Dodson,  E.  C. --Study  of  Student  Achievement  in  Mathematics. 

School  Science  and  Mathematics.  Vol. XIV. 

May  1914. 

Frailey,  L.  E. , and  Crain,  C.  M. — Correlation  of  Excellence 

in  Different  School  Subjects. 

Hancock,  H.--Crit icism  on  Mathematics  Teaching. 

School  and  Society,  Vol  VI,  September  1917. 
Lincoln,  E.  A. --Relative  Standing  in  High  School,  Early 

College  and  Entrance  Examinations. 

School  ana  Society.  Vol.V  April  1917. 

Moritz,  R.  E. --Mathemat ics  as  a Test  of  Mental  Efficiency. 

School  ana  Society  Vol.  VII,  January  1918. 


* 


. 

. 


. . 

I _ M ■ ; • «r*;'  “ 


. 


"y  i 

* 


i 


57 


Rietz  ana  Shade. --A  Study  of  University  of  Illinois  Grades. 

University  of  Illinois  Studies,  Vol.  III. 
Thorndike,  E.  L.--The  Future  of  the  College  Entrance  Examina 

tion  Board. 

Educational  Review,  Vol  XXXI,  May  1906. 
Werrmeyer,  It.  W. --Rel iao il it y of  Test  Grades  in  Mathematics. 

School  Science  and  Mathematics,  Vol.  XIV. 
May  1914. 


